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find the first derivative of sin⁡(4X) \sin(4X)

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Full solution

Q. find the first derivative of sin⁡(4X) \sin(4X)
  1. Identify Function: Identify the function to differentiate.\newlineWe are given the function f(x)=sin⁡(4x)f(x) = \sin(4x) and we need to find its first derivative, f′(x)f'(x).
  2. Apply Chain Rule: Apply the chain rule for differentiation.\newlineThe chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is sin⁡(u)\sin(u) and the inner function is u=4xu = 4x.
  3. Differentiate Outer Function: Differentiate the outer function with respect to the inner function.\newlineThe derivative of sin⁡(u)\sin(u) with respect to uu is cos⁡(u)\cos(u).
  4. Differentiate Inner Function: Differentiate the inner function with respect to xx. The derivative of 4x4x with respect to xx is 44.
  5. Apply Chain Rule Result: Apply the chain rule by multiplying the results from Step 33 and Step 44.\newlinef′(x)=cos⁡(4x)×4f'(x) = \cos(4x) \times 4
  6. Simplify Final Answer: Simplify the expression if necessary.\newlineThe expression is already simplified, so we have the final answer.\newlinef′(x)=4cos⁡(4x)f'(x) = 4\cos(4x)

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